Optimization of Peak Power in Vibrating Structures

via Semidefinite Programming

Mohammad A. Heidari

The Boeing Company, Propulsion Technology, Seattle, WA 98124

Randy Cogill∗ Pradip Sheth† Paul Allaire†

∗Department of Systems and Information Engineering†Department of Mechanical and Aerospace Engineering

University of Virginia, Charlottesville, VA 22904

In this paper we consider the problem of designing a truss to minimize peak power input

to the structure by a dynamic load. We first consider a model where known sinusoidal forces

are applied to the truss, and our goal is to choose rod cross-sectional areas to minimize

peak power subject to a constraint on the total mass of the structure. We then consider a

second model where unknown forces are applied to the truss, and our goal is to choose rod

cross-sectional areas to minimize the worst-case peak power over all unit magnitude forces.

In both cases, we show that the optimization problems can be formulated as semidefinite

programs. Numerical results are presented for both cases.

I. Background

This paper focuses on optimal design of truss structures for peak power minimization. Optimal design oftruss structures has a long history, starting with the papers5 and.6 These problems are typically formulatedin such a way that a ground structure, rod lengths, and material properties are specified, and the goal is toselect rod cross-sectional areas to optimize some objective function. The majority of existing work on thistopic involves optimization under static loads, with minimization of compliance subject to weight constraintsbeing the most common formulation.2

If the constraint set and objective function of an optimization problem is convex, the globally optimalsolution can typically be computed efficiently. Convexity of the objective function and constraint set guar-antees that the problem has a single locally optimal solution, which is the global optimum solution. So,gradient-based methods can be applied to compute the global optimum of a convex optimization problem.As is shown in,4 if the compliance minimization problem is not formulated carefully, the constraint set of theoptimization problem will not be convex. In particular, if nodal displacements are not eliminated from theproblem formulation, a non-convex bilinear constraint will be present. This lack of convexity creates multiplelocal optima, and gradient-based methods will not always compute global optima. However, by eliminatingnodal displacements, a convex optimization problem results. The paper7 gives detailed conditions underwhich optimal truss design problems can be formulated as convex optimization problems.

More recently, it has been shown in1 that structural optimization problems can be formulated as semidef-inite programs. Semidefinite programs are a generalization of linear programs which have many structuralproperties that can be exploited in their numerical solution. Specifically, semidefinite programs are problemsthat involve optimization of linear objective functions subject to the constraint that matrix variables arepositive (semi)definite. Semidefinite programming formulations of several structural optimization problemsare surveyed in.4 An additional advantage of formulating structural optimization problems as semidefiniteprograms is that there are several high performance, freely available solvers for semidefinite programs, suchas SeDuMi.8

The optimization problems that we present in this paper are dynamic extensions of the static formulationssurveyed in.4 We will show that, under certain conditions, minimization of peak power under dynamic loadsis a convex optimization problem and can be formulated as a semidefinite program. Any gradient-based

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search, such as the BIGDOT algorithm used in Nastran’s SOL 200,10 is guaranteed to compute globallyoptimal solutions to these problems due to convexity. However, we claim that formulating these problemsas semidefinite programs and using existing semidefinite programming solvers has certain advantages overusing a generic gradient method. The BIGDOT algorithm is an exterior penalty function method, meaningthat constraints are replaced with cost components in the objective function that penalize the objectivewhen a constraint is violated. This results in an unconstrained minimization problem that can be solved bystandard gradient methods. This algorithm can be applied very generally and is highly scalable to problemswith many variables and constraints. However, semidefinite programs have significant structure that can beexploited in their solution. Most existing semidefinite programming solvers are based on customized primal-dual interior-point methods, rather than generic barrier methods.9 Primal-dual algorithms seek to minimizethe gap between the primal objective value and the objective value of the dual problem. Taking this approachhas several advantages over primal algorithms, such as simple stopping criteria and simple computation ofjoint primal and dual search directions. As observed in,9 these algorithms generally outperform barriermethods when applied to semidefinite programs, typically converging in fewer than 50 iterations even forlarge problem instances. Solutions to the examples presented in this paper were computed with the SeDuMisolver,8 a freely-available semidefinite program solver for Matlab that uses primal-dual interior point methods.

II. Peak power minimization

In this section we will provide a formulation of the problem of truss design for peak power minimization.We will consider truss composed of linear elastic rods, where the rod cross sectional areas are our designvariables. For given material parameters and geometry, we can use standard finite-element methods (as in3

for example) to obtain the mass and stiffness matrices of the truss. When considering the mass and stiffnessmatrices as functions of individual rod cross-sectional areas, it turns out that the mass and stiffness matricesare both linear in the cross-sectional areas. That is, the mass and stiffness matrices of the truss can beexpressed as

M(a) = a1M1 + · · · + anMn

andK(a) = a1K1 + · · · + anKn,

where ai is the cross-sectional area of the ith rod. Our goal will be to optimize the choice of a1, . . . , an tominimize peak power delivered to the truss subject to additional constraints on the total structural mass.

For given M(a) and K(a) and a given applied force we will now provide an expression for peak power.This will be the objective that we will optimize over. Let f(t) be the vector of forces applied to nodes of thetruss, and let v(t) be the vector of nodal velocities. We define the peak steady-state power delivered to thestructure as

lim supt→∞

|f(t)T v(t)|.

Here we will consider general sinusoidal forces, which can be expressed as

f(t) = cos(ωt)fR + sin(ωt)fI

=1

2(eiωt + e−iωt)fR − i

2(eiωt − e−iωt)fI

=eiωt

2(fR − ifI) +

e−iωt

2(fR + ifI)

The nodal velocities satisfy the second-order differential equation

M(a)v′′(t) + K(a)v(t) = f ′(t)

The solution to this equation is of the form v(t) = eiωt

2(vR− ivI)+ e

−iωt

2(vR + ivI). Substituting this solution

and equating real and imaginary parts gives

(K − ω2M)vR = ωfI

(K − ω2M)vI = −ωfR.

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Power is then computed as

f(t)T v(t) =ei2ωt

4

(

(fT

RvR − fT

IvI) − i(fT

RvI + fT

IvR))

+e−i2ωt

4

(

(fT

RvR − fT

IvI) + i(fT

RvI + fT

IvR))

+1

2(fT

RvR + fT

IvI)

=1

2(fT

RvR + fT

IvI) +

1

2cos(2ωt)(fT

RvR − fT

IvI) −

1

2sin(2ωt)(fT

RvI + fT

IvR)

Assuming that (K(a) − ω2M(a))−1 exists, we can solve for vR and vI to obtain

vR = ω(K − ω2M)−1fI

vI = −ω(K − ω2M)−1fR

In our formulation we will place the stronger requirement (K(a) − ω2M(a)) ≻ 0 to guarantee invertibility.Substituting the expressions for vR and vI into the expression for power gives

f(t)T v(t) = ω(

fT

R(K − ω2M)−1fI

)

cos(2ωt) +ω

2

(

(fR + fI)T (K − ω2M)−1(fR − fI)

)

sin(2ωt)

= C sin(2ωt + θ),

where

C = ω

√

1

4

(

(fR + fI)T (K − ω2M)−1(fR − fI))2

+(

fT

R(K − ω2M)−1fI

)2

So,lim sup

t→∞|f(t)T v(t)| = C.

We will consider the problem of minimizing C subject to the constraint that

K(a) − ω2M(a) ≻ 0,

together with a constraint on the total mass of the structure. Since C is positive and C2 is monotonicallyincreasing in C, we will work with the equivalent problem of minimizing C2. If we let q be a vector ofdensities times lengths of each rod, qT a gives the mass of the structure. We would like to design a structurewith minimum peak power subject to a constraint on the total mass of the structure. If m is the maximumallowable mass, this constraint is expressed as qT a ≤ m. This leads to the peak power minimization problem:

minimize: 1

4

(

(fR + fI)T (K(a) − ω2M(a))−1(fR − fI)

)2+(

fT

R(K(a) − ω2M(a))−1fI

)2

subject to: qT a ≤ m

K(a) − ω2M(a) ≻ 0

In the next section we will discuss computational algorithms for solving this problem.

III. Semidefinite programming formulation

In this section we will consider semidefinite programming formulations of the peak power minimizationproblem. At this point it is not clear whether the the general peak power minimization problem for anyspecific applied force can be equivalently posed as a semidefinite program. What we will show are thefollowing:

• When all forces applied to the truss are in-phase, the problem of peak power minimization can be castas a semidefinite program.

• We also consider the problem of minimizing worst-case peak power over all unit-magnitude forces. Weshow that this problem has an equivalent semidefinite programming formulation. This is true evenwhen considering nodal forces that may not be in-phase.

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III.A. Peak power minimization: forces in-phase

If all applied nodal forces are in-phase, we will show that the peak power minimization problem can bereduced to a semidefinite program and solved efficiently. All applied forces are in phase if and only if thereexist some constants c1 and c2 such that c1fR = c2fI . Suppose, without loss of generality, that c2 6= 0. Thenthe objective function of the peak power minimization problem reduces to

(

1 + 2

(

c1

c2

)2

+

(

c1

c2

)4)

(

fT

R(K(a) − ω2M(a))−1fR

)2

Under the constraint that K(a) − ω2M(a) ≻ 0, minimizing this objective is equivalent to minimizing

fT

R(K(a) − ω2M(a))−1fR.

By the Schur complement formula, the conditions K(a)−ω2M(a) ≻ 0 and t > fT

R(K(a)−ω2M(a))−1fR

are equivalent to[

t fT

R

fR K(a) − ω2M(a)

]

≻ 0

So, in the case c1fR = c2fI , the problem of minimizing peak power subject to a constraint on total structuralmass is expressed as

minimize: t

subject to: qT a ≤ m[

t fT

R

fR K(a) − ω2M(a)

]

≻ 0

If t∗ is the optimal value of this semidefinite program, then ω

2t∗ is the peak power for the force f(t). In the

next section, we will show a numerical example of this approach.

III.B. Worst-case peak power minimization

The formulation discussed previously optimizes a truss for a specific applied force. More generally, wecould consider the problem of minimizing the worst-case peak power over a collection of applied forces. Forexample, we could minimize the worst-case peak power over all unit magnitude forces. That is, we couldminimize peak power over all forces satisfying fT

RfR + fT

IfI = 1. It turns out, quite surprisingly, that this

problem has a simple semidefinite programming formulation.We will start out by showing that the the worst-case force applied to any truss has all nodal forces

in-phase. To show this, suppose f∗R

and f∗I

are the worst-case force vectors applied to a truss with givencross-sectional areas a. We will show that f∗

Rand f∗

Ican always be chosen so that either f∗

R= 0 or f∗

I= 0.

We will suppose that we must have f∗T

Rf∗

R> 0 and f∗T

If∗

I> 0, and show that this leads to a contradiction.

In this case, we can define

fR =1

√

f∗T

Rf∗

R

f∗R

and

fI =1

√

f∗T

If∗

I

f∗I.

Consider the applied force

f(t) =c1

2fR(eiωt + e−iωt) − ic2

2fI(e

iωt − e−iωt)

where c2

1+ c2

2= 1. We will show that peak power is maximized with either c1 = 0 or c2 = 0. The square of

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peak power is

1

4

(

c2

1fT

R(K(a) − ω2M(a))−1fR − c2

2fT

I(K(a) − ω2M(a))−1fI

)2

+(

c1c2fT

R(K(a) − ω2M(a))−1fI

)2

≤ 1

4

(

c2

1fT

R(K(a) − ω2M(a))−1fR − c2

2fT

I(K(a) − ω2M(a))−1fI

)2

+(c2

1fT

R(K(a) − ω2M(a))−1fR)(c2

2fT

I(K(a) − ω2M(a))−1fI)

=c4

1

4

(

fT

R(K(a) − ω2M(a))−1fR

)2

+c4

2

4

(

fT

I(K(a) − ω2M(a))−1fI

)2

+c2

1c2

2

2(fT

R(K(a) − ω2M(a))−1fR)(fT

I(K(a) − ω2M(a))−1fI)

=1

4

(

c2

1fT

R(K(a) − ω2M(a))−1fR + c2

2fT

I(K(a) − ω2M(a))−1fI

)2

The inequality above follows from the Cauchy-Schwarz inequality. The choice of c1 and c2 that maximizesthis upper bound is c1 = 1 if

fT

R(K(a) − ω2M(a))−1fR ≥ fT

I(K(a) − ω2M(a))−1fI

and c2 = 1 iffT

I(K(a) − ω2M(a))−1fI > fT

R(K(a) − ω2M(a))−1fR

However, the upper bound is tight for this choice of c1 and c2. So, it is necessary that either f∗R

= 0 orf∗

I= 0.When considering the worst-case peak power delivered to a truss with given cross-sectional areas a, it is

sufficient to search for fR that satisfies fT

RfR = 1 and maximizes

1

4

(

fT

R(K(a) − ω2M(a))−1fR

)2.

We would like to find the cross sectional areas that minimize this maximum. Since fT

R(K(a)−ω2M(a))−1fR >

0, we can equivalently search for cross sectional areas that minimize the maximum value of

fT

R(K(a) − ω2M(a))−1fR

over all fR satisfying fT

RfR = 1. This is equivalent to minimizing t subject to

t > fT

R(K(a) − ω2M(a))−1fR

for all fR satisfying fT

RfR = 1. This again is equivalent to minimizing t subject to

fT

R

(

tI − (K(a) − ω2M(a))−1)

fR > 0

for all fR, or equivalentlytI − (K(a) − ω2M(a))−1 ≻ 0.

Using the Schur complement formula we arrive at the equivalent semidefinite program

minimize: t

subject to: qT a ≤ m[

tI I

I K(a) − ω2M(a)

]

≻ 0

If t∗ is the optimal value of this semidefinite program, then ω

2t∗ is the worst-case peak power.

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IV. Examples

In the following examples, we will optimize the cross sectional areas of the truss shown in Figure 1 forvarious optimization criteria. Each rod i is made from material with density ρi = 1 and Young’s modulusEi = 2.5 × 104. Vertical and horizontal rods have a length of 1 meter, and diagonal rods have length

√2

meters. In all cases we will look at applied forces with a frequency of ω = 15 rad/sec.

Figure 1. Uniform truss. All rods have equal cross-sectional areas.

IV.A. Peak power minimization: forces in phase

We will first consider the case when forces of 1√2

cos(ωt) are applied to two nodes of the truss, as shown in

Figure 2. The optimal cross sectional areas for this force are depicted by line thicknesses in Figure 2. Forthis set of cross-sectional areas, the peak power is 0.007 Watts.

It is interesting to compare the peak power of the optimal truss with that of the uniform truss, whereall rod cross-sectional areas identical. Suppose the forces 1√

2cos(ωt) is applied to the optimal and uniform

trusses, as in Figure 2. For the uniform truss, the peak power is approximately 0.022 Watts, more than threetimes the value for the optimal truss.

Figure 2. Optimal truss for the shown applied force.

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IV.B. Worst-case peak power minimization

For worst-case peak power minimization, we consider the set of all applied forces with fT

RfR + fT

IfI = 1.

Cross-sectional areas are chosen to minimize the worst-case peak power over all forces satisfying fT

RfR +

fT

IfI = 1. The optimal cross sectional areas are depicted by line thicknesses in Figure 3. For this set of

cross-sectional areas, the worst-case peak power is 0.175 Watts.We will again compare the peak power of the optimal truss with that of the uniform truss, where all

rod cross-sectional areas identical. For the uniform truss, depicted in Figure 1, the worst-case peak power is0.525 Watts, approximately three times that of the optimal truss. We can also consider the peak power for atypical applied force. Now we will consider an applied force with components that are not in phase. We willapply the forces 1

2cos(ωt) as shown by the arrows marked “A” in Figure 4 and will apply the forces 1

2sin(ωt)

as shown by the arrows marked “B”. This force is chosen to model an unbalanced rotating load. For theoptimal truss, the peak power associated with this force is approximately 0.041 Watts. For the uniformtruss, the peak power is approximately 0.099 Watts, more than twice the value for the optimal truss.

Figure 3. Optimal truss for worst-case peak power minimization.

AA

BB

Figure 4. Forces applied to the optimal worst-case design.

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References

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2M.P. Bendsoe and O. Sigmund (2004). ”Topology Optimization,” Springer, Berlin.3R. Cook, D. Malkus, M. Plesha, and R. Witt (2001). ”Concepts and Applications of Finite Element Analysis,” SIAM

Journal on Optimization, J. Wiley & Sons, New York.4E. de Klerk, C. Roos, and T. Terlaky (1995). ”Semi-definite Problems in Truss Topology Optimization,” TU Delft

Technical Report.5W. Dorn, R. Gomory, and M. Greenberg (1964). ”Automatic Design of Optimal Structures,” Journal de Mechanique,

vol. 2, pp. 25-52.6P. Fleron (1964). ”The Minimum Weight of Trusses,” Bygningsstatiske Meddelelser, vol. 35, pp. 81-96.7K. Svanberg (1984). ”On Local and Global Minima in Structural Optimization,” in New Directions in Optimum Structural

Design, Wiley, New York, pp. 327-341.8J. Sturm (2002). SeDuMi version 1.05. Available from http://fewcal.kub.nl/sturm/software/sedumi.html.9L. Vandenberghe and S. Boyd (1996). ”Semidefinite Programming,” SIAM Review, vol. 38, pp. 49-95.

10G. Vanderplaats (2002). ”Very Large Scale Optimization,” NASA Contractor Report NASA/CR-2002-211768.

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